<p>You are given an integer array <code>nums</code> and two integers, <code>k</code> and <code>limit</code>. Your task is to find a non-empty <strong><span data-keyword="subsequence-array">subsequence</span></strong> of <code>nums</code> that:</p>

<ul>
	<li>Has an <strong>alternating sum</strong> equal to <code>k</code>.</li>
	<li><strong>Maximizes</strong> the product of all its numbers <em>without the product exceeding</em> <code>limit</code>.</li>
</ul>

<p>Return the <em>product</em> of the numbers in such a subsequence. If no subsequence satisfies the requirements, return -1.</p>

<p>The <strong>alternating sum</strong> of a <strong>0-indexed</strong> array is defined as the <strong>sum</strong> of the elements at <strong>even</strong> indices <strong>minus</strong> the <strong>sum</strong> of the elements at <strong>odd</strong> indices.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [1,2,3], k = 2, limit = 10</span></p>

<p><strong>Output:</strong> <span class="example-io">6</span></p>

<p><strong>Explanation:</strong></p>

<p>The subsequences with an alternating sum of 2 are:</p>

<ul>
	<li><code>[1, 2, 3]</code>

	<ul>
		<li>Alternating Sum: <code>1 - 2 + 3 = 2</code></li>
		<li>Product: <code>1 * 2 * 3 = 6</code></li>
	</ul>
	</li>
	<li><code>[2]</code>
	<ul>
		<li>Alternating Sum: 2</li>
		<li>Product: 2</li>
	</ul>
	</li>
</ul>

<p>The maximum product within the limit is 6.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [0,2,3], k = -5, limit = 12</span></p>

<p><strong>Output:</strong> <span class="example-io">-1</span></p>

<p><strong>Explanation:</strong></p>

<p>A subsequence with an alternating sum of exactly -5 does not exist.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">nums = [2,2,3,3], k = 0, limit = 9</span></p>

<p><strong>Output:</strong> <span class="example-io">9</span></p>

<p><strong>Explanation:</strong></p>

<p>The subsequences with an alternating sum of 0 are:</p>

<ul>
	<li><code>[2, 2]</code>

	<ul>
		<li>Alternating Sum: <code>2 - 2 = 0</code></li>
		<li>Product: <code>2 * 2 = 4</code></li>
	</ul>
	</li>
	<li><code>[3, 3]</code>
	<ul>
		<li>Alternating Sum: <code>3 - 3 = 0</code></li>
		<li>Product: <code>3 * 3 = 9</code></li>
	</ul>
	</li>
	<li><code>[2, 2, 3, 3]</code>
	<ul>
		<li>Alternating Sum: <code>2 - 2 + 3 - 3 = 0</code></li>
		<li>Product: <code>2 * 2 * 3 * 3 = 36</code></li>
	</ul>
	</li>
</ul>

<p>The subsequence <code>[2, 2, 3, 3]</code> has the greatest product with an alternating sum equal to <code>k</code>, but <code>36 &gt; 9</code>. The next greatest product is 9, which is within the limit.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 150</code></li>
	<li><code>0 &lt;= nums[i] &lt;= 12</code></li>
	<li><code>-10<sup>5</sup> &lt;= k &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= limit &lt;= 5000</code></li>
</ul>
